Alternating Tensor

Relevant parts to questions…

• Using the definition that ${ϵ}_{ijk}=\{\begin{array}{ll}0& \text{if any of }i,j,k\text{ are equal}\\ +1& \text{if }(i,j,k)=(1,2,3),(2,3,1),\text{or }(3,1,2)\\ -1& \text{if }(i,j,k)=(1,3,2),(2,1,3),\text{or }(3,2,1)\end{array}$, i.e., 1 if $(i,j,k)$ is an even permutation of $(1,2,3)$, -1 if it’s an odd permutation, or 0 if any.
• Using $\mathbf{a}\times \mathbf{b}={ϵ}_{ijk}{a}_j{b}_k$.
• Using the property of ${ϵ}_{ijk}$ that you can swap two indices to switch its polarity between $+1$ and $-1$.

The Alternating Tensor is defined as ${ϵ}_{ijk}=\{\begin{array}{ll}0& \text{if any of }i,j,k\text{ are equal}\\ +1& \text{if }(i,j,k)=(1,2,3),(2,3,1),\text{or }(3,1,2)\\ -1& \text{if }(i,j,k)=(1,3,2),(2,1,3),\text{or }(3,2,1)\end{array}$, i.e., 1 if $(i,j,k)$ is an even permutation of $(1,2,3)$, -1 if it’s an odd permutation, or 0 if any.

By this definition, the Alternating Tensor has the following properties:

• ${ϵ}_{ijk}$ is unchanged if indices are reordered by a cyclic permutation, i.e., ${ϵ}_{ijk}={ϵ}_{jki}={ϵ}_{kij}$.
• ${ϵ}_{ijk}$ is changed if any two of the suffices are interchanged, i.e., ${ϵ}_{ijk}=-{ϵ}_{jik}$; that is, the alternating tensor is anti-symmetric.

The Alternating Tensor can then be used to write multiple operations, for instance:

1. Cross product of two vectors, $\mathbf{a}\times \mathbf{b}=\left|\begin{array}{ccc}\mathbf{i}& \mathbf{j}& \mathbf{k}\\ a_1& a_2& a_3\\ b_1& b_2& b_3\end{array}\right|=\left|\begin{array}{cc}{a}_2& a_3\\ b_2& b_3\end{array}\right|\mathbf{i}-\left|\begin{array}{cc}{a}_1& a_3\\ b_1& b_3\end{array}\right|\mathbf{j}+\left|\begin{array}{cc}{a}_1& a_2\\ b_1& b_2\end{array}\right|\mathbf{k}$, using the Alternating Tensor::$(\mathbf{a}\times \mathbf{b}{)}_i={ϵ}_{ijk}{a}_j{b}_k$.
2. Determinant of $3\times 3$ matrices, using the Alternating Tensor::${ϵ}_{pqr}|M|={ϵ}_{ijk}{M}_{pi}{M}_{qj}{M}_{rk}$, which can be simplified by using either rows or columns.
3. The scalar triple product, $\mathbf{a}\cdot (\mathbf{b}\times \mathbf{c})$, using the Alternating Tensor::$\mathbf{a}\cdot (\mathbf{b}\times \mathbf{c})={ϵ}_{ijk}{a}_i{b}_j{c}_k$.